On wavelet-based features for classifying mass spectrometry data

The problem in hand

Our data is a mass spectrum signal data (functional data).

• \(p >90000\) features or dimensions and \(p <100\) subjects or observations

• Our goal is develop a methodology to classify the functional data, such as cancer mass spectrum data (Cancer is the second leading cause of death in the world after heart disease)
• Our focus is to leverage the capabilities of Wavelet Analysis.

Wavelet Analysis

The Fourier Transform of a signal \(x(t)\) can be expressed as:

\[ X(f)= \int_{-\infty}^{\infty} x(t) e^{i2 \pi ft} dt \tag{2} \](\(e^{ix}= \cos x + i \sin x\), Euler’s formula); \(f\) is the frequency domain.

The Wavelet Transform of a signal \(x(t)\) can be given as:

\[ WT(s,\tau)= \frac{1}{\sqrt s}\int_{-\infty}^{\infty} x(t) \psi^*\big(\frac{t-\tau}{s}\big) dt, \tag{3} \]

where \(\psi^*(t)\) denotes the complex conjugate of the base wavelet \(\psi(t)\)); \(s\) is the scaling parameter, and \(\tau\) is the location parameter.

Example: Morlet Wavelet \(\psi(t) = e^{i2 \pi f_0t} e^{-(\alpha t^2/\beta^2)}\), with the parameters \(f_0\), \(\alpha\), \(\beta\) all being constants.

Wavelet Families

Discrete Wavelet Transform

Related Work

• Ovarian cancer detection (Yu et al. 2005): A combination of Binning, Kolmogorov-Smirnov test, discrete wavelet transform, and support vector machines.

• Proteomic profile with bi-orthogonal discrete wavelet transform (Schleif et al. 2009): A combination of outlier detection-centroied-based, recalibration, baseline correction (top-hat filter), Kolmogorov-Smirnov test, discrete wavelet transform bior 3.7, and support vector machines.

• Ovarian cancer detection using peaks and discrete wavelet transform (Du et al. 2009): A combination of discrete wavelet transform, thresholding, peak detection using MAD, Kolmogorov-Smirnov test, bagging predictor.

Related Work

• Ovarian cancer classification using wavelets and genetic algorithm (Nguyen et al. 2015): A combination of Haar discrete wavelet transform, genetic algorithms.

• Breat cancer mass spectrum classification (Cohen, Messaoudi, and Badir 2018): A combination of segmentation, discrete wavelet transform, statistical features on the coefficients, PCA-T2 Hotelling statistic, SVM.

• Ovarian cancer mass spectrum classification (Vimalajeewa, Bruce, and Vidakovic 2023): A combination of Daubechies-7 wavelet transform, sample variance and distance variance, Fisher’s criterion for feature extraction, SVM, KNN, and Logistic regression.

Methods

Variables of the experimental design:

• Four pre-processing techniques.

  • 5 window sizes.

  • Two are wavelet-based and two are not.

  • 10 wavelets families

• Four ML Models: Logistic Regression, Support Vector Machine, Random Forest, and XGboost.

• Two sampling: up and no sampling to overcome the imbalance classes

A total of 8800 models were run.

Methods

• Processing 1 (PROC1): The feature space includes mean, variance, energy, coefficient of variation, Skewness, and Kurtosis; wavelet transform.

• Processing 2 (PROC2): Same as PROC1 but the feature space will include the first 10 autocorrelation coefficients.

• Processing 3 (PROC3): Same as PROC1 but without the wavelet transform.

• Processing 4 (PROC4): Same as PROC2 but without the wavelet transform.

Methods

The PCA-Hotelling \(T^2\) statistic

Consider a normalized data matrix, with \(p\) variables and \(N\) observations:

\[ \textbf{Z}= \left(\begin{array}{cccc}z_{11} & z_{12} & \dots &z_{1p}\\ z_{21} & z_{22} & \dots &z_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ z_{N1} & z_{N2} & \dots & z_{Np}\end{array}\right) \]

The covariance matrix of \(Z\) can be approximated as:

\[ S= \frac{1}{N-1}Z^{T}Z= P \Lambda P^{T} \]

where \(\Lambda= diag (\lambda_1, \lambda_2, ..., \lambda_p)\) with \(\lambda_1 \ge \lambda_2 \ge ..., \ge \lambda_p\). \(\lambda_i\) are the eigenvalues and P are the eigenvectors of \(S\).

Methods

The PCA-Hotelling \(T^2\) statistic

According to \(\lambda_i\)’s, \(P\) and \(\Lambda\) could be divided into a feature space () and a residual space (). We can then rewrite \(P\) and \(\Lambda\) as follows:

\[ P= \left[\begin{array}{cc} P_{feat} & P_{res}\end{array}\right] \]

\[ \Lambda= \left[\begin{array}{cc}\Lambda_{feat} & 0 \\ 0 & \Lambda_{res}\end{array}\right] \]

The Hotelling \(T^2\) statistic can then be computed as follows:

\[ {T^2}= Z P_{feat} \Lambda^{-1}_{feat} P^T_{feat} Z^{T} \tag{4} \]

where \(T^2\) is the Hotelling statistic calculated into the multivariate feature space of the principal component analysis, and \(P^T\) is the transpose of \(P\).

Results

PROC3 seems to have the highest values across all models and metrics.

Results

The larger the sample size, the higher the average accuracy. (not controlling for other variables)

Results

Conclusions

• The wavelet coefficients do not always lead to improved performance.

• Additionally, autocorrelations do not appear to be effective predictors, which is anticipated for wavelet coefficients.

Future work

• Enhance the evaluation using Monte Carlo simulation.
• Identify new datasets to test the methodology.
• Incorporate additional wavelet families.

Thank you

• Contact: M.C.: mjc106@students.uwf.edu and A.C.: acohen@uwf.edu
• Announcement: Special Issue on Statistical Monitoring and AI Models. Deadline June 2025.